Renyi entropy of the XY spin chain

We consider the one-dimensional XY quantum spin chain in a transverse magnetic field. We are interested in the Renyi entropy of a block of L neighboring spins at zero temperature on an infinite lattice. The Renyi entropy is essentially the trace of some power α of the density matrix of the block. We calculate the asymptotic for L → ∞ analytically in terms of Klein's elliptic λ-function. We study the limiting entropy as a function of its parameter α. We show that up to the trivial addition terms and multiplicative factors, and after a proper rescaling, the Renyi entropy is an automorphic function with respect to a certain subgroup of the modular group; moreover, the subgroup depends on whether the magnetic field is above or below its critical value. Using this fact, we derive the transformation properties of the Renyi entropy under the map α → α^-1 and show that the entropy becomes an elementary function of the magnetic field and the anisotropy when α is an integer power of 2; this includes the purity tr ρ². We also analyze the behavior of the entropy as α → 0 and ∞ and at the critical magnetic field and in the isotropic limit (XX model).